Conical-vertex conjecture for extremal uniform set systems

For integers $r\geq 3$, $t\geq r-2$, and $m\geq 0$, let $s_{r,t}(m)$ be the maximum size of the family in an $(r,t)$-system $(H,\mathcal{F})$ with $|H|=m$. An extremal system is one attaining this maximum, and a vertex is conical if it is adjacent to every other vertex of its underlying graph. Set-system conical-vertex conjecture. For all integers $r\geq 4$ and $t\geq r-2$, every underlying graph of an extremal system for $s_{r,t}(m)$ has a conical vertex for large enough $m$. This predicts that extremal systems for the set-function problem inherit the recursive conical structure found in the graph formulation.